Product of Absolute Values on Ordered Integral Domain

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Theorem

Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is denoted by $0_D$.

For all $a \in D$, let $\size a$ denote the absolute value of $a$.


Then:

$\size a \times \size b = \size {a \times b}$


Proof

Let $P$ be the (strict) positivity property on $D$.

Let $<$ be the (strict) total ordering defined on $D$ as:

$a < b \iff a \le b \land a \ne b$

Let $N$ be the strict negativity property on $D$.


We consider all possibilities in turn.


$(1): \quad a = 0_D$ or $b = 0_D$

In this case, both the left hand side $\size a \times \size b$ and the right hand side are equal to zero.

So:

$\size a \times \size b = \size {a \times b}$


$(2): \quad \map P a, \map P b$

First:

\(\displaystyle \map P a, \map P b\) \(\leadsto\) \(\displaystyle \size a = a, \size b = b\) Definition of Absolute Value on Ordered Integral Domain
\(\displaystyle \) \(\leadsto\) \(\displaystyle \size a \times \size b = a \times b\)

Then:

\(\displaystyle \map P a, \map P b\) \(\leadsto\) \(\displaystyle \map P {a \times b}\) Strict Positivity Property: $(P \, 2)$
\(\displaystyle \) \(\leadsto\) \(\displaystyle \size {a \times b} = a \times b\) Definition of Absolute Value on Ordered Integral Domain

So:

$\size a \times \size b = \size {a \times b}$


$(3): \quad \map P a, \map N b$

First:

\(\displaystyle \map P a, \map N b\) \(\leadsto\) \(\displaystyle \size a = a, \size b = -b\) Definition of Absolute Value on Ordered Integral Domain
\(\displaystyle \) \(\leadsto\) \(\displaystyle \size a \times \size b = -a \times b\) Product with Ring Negative

Then:

\(\displaystyle \map P a, \map N b\) \(\leadsto\) \(\displaystyle \map N {a \times b}\) Properties of Strict Negativity: $(5)$
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map P {-a \times b}\) Definition of Strict Negativity Property
\(\displaystyle \) \(\leadsto\) \(\displaystyle \size {a \times b} = -a \times b\) Definition of Absolute Value on Ordered Integral Domain

So:

$\size a \times \size b = \size {a \times b}$

Similarly $\map N a, \map P b$.


$(4): \quad \map N a, \map N b$

First:

\(\displaystyle \map N a, \map N b\) \(\leadsto\) \(\displaystyle \size a = -a, \size b = -b\) Definition of Absolute Value on Ordered Integral Domain
\(\displaystyle \) \(\leadsto\) \(\displaystyle \size a \times \size b = a \times b\) Product of Ring Negatives

Then:

\(\displaystyle \map N a, \map N b\) \(\leadsto\) \(\displaystyle \map P {a \times b}\) Properties of Strict Negativity: $(4)$
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map P {a \times b}\) Definition of Strict Negativity Property
\(\displaystyle \) \(\leadsto\) \(\displaystyle \size {a \times b} = a \times b\) Definition of Absolute Value on Ordered Integral Domain

So:

$\size a \times \size b = \size {a \times b}$


In all cases the result holds.

$\blacksquare$


Sources